Marcelo Bielsa's men managed to comeback twice from a goal down to secure a spot in the final four, dashing the hopes of the Germans, who looked poised to comeback.
After a 4-2 win in Germany last week, Marcelo Bielsa's men managed to comeback from a goal down on two occassions to level the score and secure the tie, setting up a date with Portugal's Sporting Lisbon in the last four.
In this one, it took just 10 minutes for the visitors to create their first chance. A lobbed ball into the box found Huntelaar, who acrobatically volleyed it on frame, but Gorka Iraizoz managed to get a hand on it, sending it over the goal.
Both teams managed to advance up the pitch in the ensuing minutes but failed to create quality threats until the German's Dutch dynamo rebounded to fire home the first.
In the 28th minute, Huntelaar gained possession of the ball at the top of the box after a pass from Obasi, before moving toward the center and firing a low, right-footed effort into the lower left of the net, beating the outstretched hand of the Bilbao goalkeeper.
Needing two more goals to take the advantage in the tie, a free kick from Huntelaar flew over the net from 30 yards just minutes later. The wasted chance would prove costly.
Ander Herrera was taken off with an injury a half an hour in, with Ibai Gomez replacing him. The fresh face managed to turn the tide just five minutes from the break after receiving the ball on the left wing. Gomez took a touch inside from well out before hitting a perfect, world-class effort that curled past Lars Unnerstall, falling into the inside of the far post to make it 1-1.
Needing three more goals and none from the hosts to advance, Schalke failed to get another one before the half ended.
In the second period, the guests got one back with a delightful strike from the top of the box. After receiving a ball from Obasi, Raul drilled a curling effort into the right side of the goal, which hit off the post, making it 2-1 and providing a bit of hope for his side.
Just moments later, Marcelo Bielsa's man got one back, scoring the 10th goal in the tie, with Markel Susaeta, who finished with a low touch from the right side of the box into the right side of the goal after a through ball from Fernando Llorente.
Bilbao aimed to keep its advantage with defensive play for the remainder of the match while Schalke looked to secure an improbable turn around, but the home side did enough to hold on and advance.
The winners will now play the semi-finals on April 19 and 26, while Huub Stevens' men see their hope of European glory reach its end this season.